Daniel H. Rothman

Immiscible cellular-automaton fluids

We introduce a new deterministic collision rule for lattice-gas (cellular-automaton) hydrodynamics that yields immiscible two-phase flow. The rule is based on a minimization principle and the conservation of mass, momentum, and particle type. A numerical example demonstrates the spontaneous separation of two phases in two dimensions. Numerical studies show that the surface tension coefficient obeys Laplaces formula.

Calibrating the End-Permian Mass Extinction

High-precision geochronologic dating constrains probable causes of Earths largest mass extinction. The end-Permian mass extinction was the most severe biodiversity crisis in Earth history. To better constrain the timing, and ultimately the causes of this event, we collected a suite of geochronologic, isotopic, and biostratigraphic data on several well-preserved sedimentary sections in South China. High-precision U-Pb dating reveals that the extinction peak occurred just before 252.28 ± 0.08 million years ago, after a decline of 2 per mil (‰) in δ13C over 90,000 years, and coincided with a δ13C excursion of −5‰ that is estimated to have lasted ≤20,000 years. The extinction interval was less than 200,000 years and synchronous in marine and terrestrial realms; associated charcoal-rich and soot-bearing layers indicate widespread wildfires on land. A massive release of thermogenic carbon dioxide and/or methane may have caused the catastrophic extinction.

Dynamics of the Neoproterozoic carbon cycle

The existence of unusually large fluctuations in the Neoproterozoic (1,000–543 million years ago) carbon-isotopic record implies strong perturbations to the Earths carbon cycle. To analyze these fluctuations, we examine records of both the isotopic content of carbonate carbon and the fractionation between carbonate and marine organic carbon. Together, these are inconsistent with conventional, steady-state models of the carbon cycle. The records can be well understood, however, as deriving from the nonsteady dynamics of two reactive pools of carbon. The lack of a steady state is traced to an unusually large oceanic reservoir of organic carbon. We suggest that the most significant of the Neoproterozoic negative carbon-isotopic excursions resulted from increased remineralization of this reservoir. The terminal event, at the Proterozoic–Cambrian boundary, signals the final diminution of the reservoir, a process that was likely initiated by evolutionary innovations that increased export of organic matter to the deep sea.

Lattice-gas cellular automata : simple models of complex hydrodynamics

Preface Acknowledgements 1. A simple model of fluid mechanics 2. Two routes to hydrodynamics 3. Inviscid two-dimensional lattice-gas hydrodynamics 4. Viscous two-dimensional hydrodynamics 5. Some simple 3D models 6. The lattice-Boltzmann method 7. Using the Boltzmann method 8. Miscible fluids 9. Immiscible lattice gases 10. Lattice-Boltzmann method for immiscible fluids 11. Immiscible lattice gases in three dimensions 12. Liquid-gas models 13. Flow through porous media 14. Equilibrium statistical mechanics 15. Hydrodynamics in the Boltzmann approximation 16. Phase separation 17. Interfaces 18. Complex fluids and patterns Appendices Author Index Subject Index.

Nonlinear inversion, statistical mechanics, and residual statics estimation

Nonlinear inverse problems are usually solved with linearized techniques that depend strongly on the accuracy of initial estimates of the model parameters. With linearization, objective functions can be minimized efficiently, but the risk of local rather than global optimization can be severe. I address the problem confronted in nonlinear inversion when no good initial guess of the model parameters can be made. The fully nonlinear approach presented is rooted in statistical mechanics. Although a large nonlinear problem might appear computationally intractable without linearization, reformulation of the same problem into smaller, interdependent parts can lead to tractable computation while preserving nonlinearities. I formulate inversion as a problem of Bayesian estimation, in which the prior probability distribution is the Gibbs distribution of statistical mechanics. Solutions are then obtained by maximizing the posterior probability of the model parameters. Optimization is performed with a Monte Carlo te...

Cellular-automaton fluids; a model for flow in porous media

Numerical models of fluid flow through porous media can be developed from either microscopic or macroscopic properties. The large‐scale viewpoint is perhaps the most prevalent. Darcy’s law relates the chief macroscopic parameters of interest—flow rate, permeability, viscosity, and pressure gradient—and may be invoked to solve for any of these parameters when the others are known. In practical situations, however, this solution may not be possible. Attention is then typically focused on the estimation of permeability, and numerous numerical methods based on knowledge of the microscopic pore‐space geometry have been proposed. Because the intrinsic inhomogeneity of porous media makes the application of proper boundary conditions difficult, microscopic flow calculations have typically been achieved with idealized arrays of geometrically simple pores, throats, and cracks. I propose here an attractive alternative which can freely and accurately model fluid flow in grossly irregular geometries. This new method s...

Automatic estimation of large residual statics corrections

Conventional approaches to residual statics estimation obtain solutions by performing linear inversion of observed traveltime deviations. A crucial component of these procedures is picking time delays; gross errors in these picks are known as “cycle skips” or “leg jumps” and are the bane of linear traveltime inversion schemes. This paper augments Rothman (1985), which demonstrated that the estimation of large statics in noise‐contaminated data is posed better as a nonlinear, rather than as a linear, inverse problem. Cycle skips then appear as local (secondary) minima of the resulting nonlinear optimization problem. In the earlier paper, a Monte Carlo technique from statistical mechanics was adapted to perform global optimization, and the technique was applied to synthetic data. Here I present an application of a similar Monte Carlo method to field data from the Wyoming Overthrust belt. Key changes, however, have led to a more efficient and practical algorithm. The new technique performs explicit crosscorr...

Transport in sandstone: A study based on three dimensional microtomography

High resolution imaging of the microstructure of Fontainebleau sandstone allows a direct comparison between theoretical calculations and laboratory measurements. While porosity, pore-volume-to-surface ratio, permeability, and end point relative permeability are well predicted by our calculations, we find that electrical resistivity and wetting phase residual saturation are both overestimated.

Lattice-Boltzmann simulations of flow through Fontainebleau sandstone

We report preliminary results from simulations of single-phase and two-phase flow through three-dimensional tomographic reconstructions of Fontainebleau sandstone. The simulations are performed with the lattice-Boltzmann method, a variant of lattice-gas cellular-automation models of fluid mechanics. Simulations of single-phase flow on a sample of linear size 0.2 cm yield a calculated permeability in the range 1.0–1.5 darcys, depending on direction, which compares qualitatively well with a laboratory measurement of 1.3 darcys on a sample approximately an order of magnitude larger. The sensitivity of permeability calculations to sample size, grid resolution, and choice of model parameters is quantified empirically. We also present a qualitative study of immiscible two-phase flow in a sample of linear size 0.05 cm; simulations of both drainage and imbibition are presented.

Lattice‐Boltzmann studies of immiscible two‐phase flow through porous media

Using a recently introduced numerical technique known as a lattice-Boltzmann method, we numerically investigate immiscible two-phase flow in a three-dimensional microscopic model of a porous medium and attempt to establish the form of the macroscopic flow law. We observe that the conventional linear description of the flow is applicable for high levels of forcing when the relative effects of capillary forces are small. However, at low levels of forcing capillary effects become important and the flow law becomes nonlinear. By constructing a two-dimensional phase diagram in the parameter space of nonwetting saturation and dimensionless forcing, we delineate the various regions of linearity and nonlinearity and attempt to explain the underlying physical mechanisms that create these regions. In particular, we show that the appearance of percolated, throughgoing flow paths depends not only on the relative concentrations of the two fluids but also on a dimensionless number that represents the ratio of the applied force to the force necessary for nonwetting fluid to fully penetrate through the porous medium. Finally, we fit a linear model to our simulation data in the high-forcing regions of the system and observe that an Onsager reciprocity holds for the viscous coupling of the two fluids.